Decimals More or Less!

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1.1

1

Decimals—More or Less!

I

n this unit you will consider everyday situations in which decimals are used. For example, you can measure a person’s height using inches or centimeters, and then record the results as a decimal. So, 66 inches is 5.5 feet, and 102 cm is 1.02 meters. You can measure the time it takes a person to run a race, and then record the results. You can also use decimals to find the cost of items that you buy at the store.

About How Much?

W

hen you are working with decimals, it is helpful to use what you know about fractions. Here is a number line labeled with some of the fraction and decimal benchmarks you learned about in Bits and Pieces I.

These benchmarks are useful when you are estimating with decimals. For example, you can use a benchmark to quickly estimate the total amount of a bill.

0

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

1 2

14 ¹₂ ³₄ 1¹₄ 1¹₂ 1³₄

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Problem 1.1

Tat Ming estimates total cost as he adds items to his cart in the grocery store. He wants to make sure he has enough money to pay the cashier. He puts the following items in his cart:

Chips $2.79 Salsa $1.99 Ground Beef $3.12

Cheese $1.29 Jalapeños $0.45

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Estimate the total cost and tell what you think he might be making for his friends!

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Tat Ming has only $10.00. From your estimate, does he have enough money? How confident are you of your answer?

As you work with decimals in this unit, estimate before you start the calculations. This will help you to know what answer to expect. If your estimate and your answer are not close, you may have made a mistake in calculating, even if you are using a calculator.

Estimating With Decimals

For each situation decide which operation to use. Then use benchmarks and other strategies to estimate the sum or difference.

A. Nick is going to Big Thrifty Acres to spend the $20 he got for a birthday present. His mom offers to pay the sales tax for him. He cannot spend more than $20. As he walks through the store, he has to estimate the total cost of all the items he wants to buy.

Getting Ready for Problem

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2. Nick finds some other things he would like to have. He finds a CD on sale for $5.99, a package of basketball cards for $2.89, a bag of peanuts for $1.59, and a baseball hat for $4.29. Does Nick have enough money to buy everything he wants?

3. In this situation, would you overestimate or underestimate? Why?

4. Nick decides to spend $10 and save the rest for another time. What can Nick buy from the items he wants to come as close as possible to spending $10?

B. Maria is saving to buy a new bicycle. The price for the bike she wants is $129.89. She has saved $78 from babysitting. She owes her brother

$5. Her grandmother gives her $25 for her birthday. She expects another $10 or $12 from babysitting this weekend. She empties her piggy bank and finds $13.73. Should she plan to buy the bike next week? Why or why not?

C. What strategies do you find useful in estimating sums and differences with decimals?

Homework starts on page 13.

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1.2

1.2 Adding and Subtracting Decimals P

erhaps something like the following happened to you.

Sally Jane and her friend Zeke buy snacks at Quick Shop. They pick out a bag of pretzels for $0.89 and a half-gallon of cider for $1.97. The cash register in the express lane is broken and the clerk says the bill (before taxes) is $10.87.

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Do you agree? If not, explain what the clerk probably did wrong.

It looks like the Quick Shop clerk does not know about place value! For example, in the numbers 236.5 and 23.65, the 2, 3, 6, and 5 mean different things because the decimal point is in a different place. The chart below shows the two numbers and the place value for each digit.

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Why is it an error for the clerk to add the 8 in the price of the pretzels to the 1 in the price of the cider?

Here is a situation that uses decimals but does

3 6 5

Tenths Hundr

edths

Thousandths Ones

2

Tens Hundr

eds

Thousands

6 5

Tenths Hundr

edths

Thousandths Ones

3 Tens 2

Hundr eds

Thousands

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Problem 1.2 Adding and Subtracting Decimals

A. Solve each problem. Write a mathematical sentence using decimal notation to show your computation. Record your sentence in a table like the one below. You will add to your table in Problem 1.3.

1. Carmela signed up to clean 1.5 miles for the cross-country team.

It starts to rain after she has cleaned 0.25 of a mile. How much does she have left to clean?

2. Pam cleans 0.25 of a mile for the chorus and cleans another 0.375 of a mile for the math club. How much does she clean altogether?

3. Jim, a member of the chess club, first cleans 0.287 of a mile. He later cleans another 0.02 of a mile. How much of a mile does he clean altogether?

4. Teri doesn’t notice that she finished her section of highway until she is 0.005 of a mile past her goal of 0.85 of a mile. She claims she cleaned nine tenths of a mile. Is she correct? Explain.

B. 1. Explain what place value has to do with adding and subtracting decimals.

2. Use your ideas about place value and adding and subtracting decimals to solve the following problems.

a. 27.9 + 103.2 b. 0.45 + 1.2

c. 2.011 + 1.99 ^d. 34.023 - 1.23 e. 4.32 - 1.746 ^f. 0.982 - 0.2 Homework starts on page 13.

Number Sentence (decimal notation) Carmela

(Leave this column blank for Problem 1.3) Person

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Problem 1.3

1.3 Using Fractions to Add and Subtract Decimals T

o add or subtract decimals, you have to be sure to only add or subtract

digits that have the same place value. You can make sure by writing addition and subtraction problems in column form and lining up the decimal points.

You can use your knowledge of fractions to see why this strategy for adding or subtracting decimals works. Remember that decimals can be written as fractions with 10; 100; 1,000; 10,000; etc. as denominators. Revisit the Quick Shop problem and think of the money amounts as fractions.

Remember that Sally Jane and her friend Zeke went to Quick Shop to buy snacks. They picked out a bag of pretzels for $0.89 and a half-gallon of cider for $1.97.

0.89 = 1.97 =

So the total cost is + = = 2.86.

How is this like thinking of the cost in pennies and then finally writing the sum in dollars?

Using Fractions to Add and Subtract Decimals

286 100 197 100 89

100

197100 89 100

1.3

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3. Jim, a member of the chess club, first cleans 0.287 of a mile. He later cleans another 0.02 of a mile. How much of a mile does he clean altogether?

4. Teri doesn’t notice that she finished her section of highway until she is 0.005 of a mile past her goal of 0.85 of a mile. She claims she cleaned nine tenths of a mile. Is she correct? Explain.

B. Use your table to compare your sentences in Problem 1.3A to those you wrote in Problem 1.2A. How does the fraction method help explain why you can line up the decimals and add digits with the same place values to find the answer?

C. Fraction benchmarks are a useful way to estimate in decimal situations.

For parts (1)–(6), write a number sentence using fraction benchmarks to estimate the sum or difference.

1. 1.199 + 2.02 2. 1.762 + 6.9 3. 0.243 + 0.7 4. 3.724 - 0.49 5. 6.899 - 2.9 6. 7.5097 - 1.008

Homework starts on page 13.

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Problem 1.4

1.4 Decimal Sum and Difference Algorithms Y

ou have looked at how place value and fraction addition and subtraction can help you make sense of adding and subtracting decimals. You can use those ideas to describe an algorithm for adding and subtracting decimals.

Decimal Sum and Difference Algorithms

A. Use your experiences adding and subtracting decimals in money and measurement situations. Describe an algorithm for adding and subtracting decimal numbers.

B. In Bits and Pieces II, you learned about fact families. Here is an addition-subtraction fact family that uses fractions:

+ = + = - = - =

1. Write the complete addition-subtraction fact family for 0.02 + 0.103 = 0.123.

2. Write the complete addition-subtraction fact family for 1.82 - 0.103 = 1.717.

C. Find the value of N that makes the mathematical sentence correct.

Fact families might help.

1. 63.2 + 21.075 = N ^2. 44.32 - 4.02 = N 3. N + 2.3 = 6.55 4. N - 6.88 = 7.21

D. 1. Explain how you can solve the problem 4.27 - by changing to a decimal.

2. Explain how you can solve the problem 4.27 - 2¹₈ by changing 4.27 2¹₈

2¹₈

12 13 5 1 6

1 3 2 5 6 5

1 6 1 2 3 5

1 6 1 3 2